System and method for super-resolution imaging from a sequence of color filter array (CFA) low-resolution images

ABSTRACT

A method and system for improving picture quality of color images by combing the content of a plurality of frames of the same subject; comprising: at least one processor; the at least one processor comprising a memory for storing a plurality of frames of a subject; the at least one processor operating to combine the content of plurality of frames of the subject into a combined color image by performing: a process in which at least two multicolored frames are converted to monochromatic predetermined color frames; a gross shift process in which the gross shift translation of one monochromatic predetermined color frame is determined relative to a reference monochromatic predetermined color frame; a subpixel shift process utilizing a correlation method to determine the translational and/or rotational differences of one monochromatic predetermined color frame to the reference monochromatic predetermined color frame to estimate sub-pixel shifts and/or rotations between the frames; and an error reduction process to determine whether the resolution of the resulting combined color image is of sufficient resolution; the error reduction process comprising applying at least one spatial frequency domain constraint and at least one spatial domain constraint to the combined color image to produce at least one high-resolution full color image.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation-In-Part of U.S. patent applicationSer. No. 12/576,132, filed Oct. 8, 2009, entitled System and Method ofSuper-Resolution Imaging from a Sequence of Translated and RotatedLow-Resolution Images, and which in turn is a Continuation-In-Part ofapplication Ser. No. 11/038,401, filed on Jan. 19, 2005, now U.S. Pat.No. 7,602,997, entitled “Method of Super-Resolving Images.” Priority isbeing claimed under 35 U.S.C. §120 to both application Ser. Nos.12/576,132 and 11/038,401, and both are hereby incorporated byreference.

GOVERNMENT INTEREST

The invention described herein may be manufactured, used, and/orlicensed by or for the United States Government.

FIELD OF THE INVENTION

This invention relates in general to a method of digital imageprocessing, and more specifically relates to super-resolution imagereconstruction.

BACKGROUND OF THE INVENTION

The terminology “subpixel” or “subpixels” as used herein refers to oneor more components of a single pixel. In a color subpixelated image, thecomponents of a single pixel comprise several primary color primaries,which may be three ordered color elements such as blue, green, and red(BGR), or red, green, and blue (RGB), or green, blue, red. Some imagedisplays have more than three primaries, further including yellow(RGBY), or white (RGBW), or yellow and cyan (RGBYC). Sub-pixels mayappear as a single color when viewed directly by the human eye due tospatial integration by the eye. However, the components may be visibleunder a magnification. Upon achieving a predetermined resolution, thecolors in the sub-pixels are not visible, but the relative intensity ofthe components may cause a shift in the apparent position or orientationof a line. The resolution at which sub-pixels go unnoticed may dependupon the viewer or the application as the visual processes of humansvary; whereby the colored “fringes” resulting from sub-pixel renderingmay cause distraction in some individuals or applications but notothers. Methods that account for sub-pixel rendering are generallycalled subpixel rendering algorithms.

Image resolution relates to the detail that an image possesses. Forsatellite images, the resolution generally correlates to the arearepresented by each pixel. Generally speaking, an image is considered tobe more accurate and detailed as the area represented by each pixel isdecreased. As used herein, the term images includes digital images, filmimages, and/or other types of images.

As used herein, the terminology “sampling” refers to the practice ofcollecting a subset of individual observations intended to yield someknowledge about a subject. “Sampling” includes the act, process, ortechnique of selecting a representative part of a population or sequenceof moving images for the purpose of determining parameters orcharacteristics of the whole population or moving image sequence. Forexample, according to the 1949 sampling theorem by C E Shannon, in orderto recover a signal function ƒ(t) precisely, the sampling rate must bedone at a rate greater than twice the signal's highest frequencycomponent.

Many low-cost sensors (or cameras) may spatially or electronicallyundersample an image. Similarly, cameras taking pictures from greatdistances, such as aerial photos, may not obtain detailed informationabout the subject matter. This may result in aliased images in which thehigh frequency components are folded into the low frequency componentsin the image. Consequently, subtle or detail information (high frequencycomponents) are not present in the images.

Many digital cameras acquire images using a single image sensor overlaidwith a color filter array (CFA), and demosaicing is required to renderthese images into a viewable format. “Demosaicing process” is a digitalimage process of reconstructing a full color image from incomplete colorsamples output from an image sensor or camera overlaid with a colorfilter array (CFA).

When an image is captured by a monochrome camera, a singlecharge-coupled device (CCD) or complementary metal-oxide semiconductor(CMOS) sensor is used to sample the light intensity projected onto thesensor. Color images are captured in much the same way, except that thelight intensity is measured in separate color channels, usually red,green, and blue. In order to do this, three separate sensors could beused in conjunction with a beam splitter to accurately measure each ofthe three primary colors at each pixel. However, this approach isexpensive and mechanically difficult to implement, making its use incommercial imaging systems infeasible. To overcome this obstacle, thecolor filter array (CFA) was introduced to capture a color image usingonly one sensor.

A CFA is an array that is used in front of the image sensors toalternate color filters that samples only one color channel at eachpixel location. The most popular and common CFA is arranged in mosaicpattern, called the Bayer pattern, as described in U.S. Pat. No.3,971,065, “Color image array,” July 1976, B. E. Bayer, which isillustrated in FIG. 1. For each 2×2 set of pixels, two diagonallyopposed pixels have green filters, and the other two pixels have red andblue filters, which is called quincunx sampling pattern. A Bayer imagecan be also seen as a grayscale image as shown in FIG. 28.

This pattern results in half of the image resolution being dedicated toaccurate measurement of the green color channel and quarter of the imageresolution of the red or blue color channel. The peak sensitivity of thehuman visual system lies in the medium wavelengths, justifying the extragreen sampling as described in X. Li, B. Gunturk, and L. Zhang, “Imagedemosaicing: a systematic survey,” Proc. SPIE, Vol. 6822, 68221J, 2008.Because each pixel now has only one color sampled, in order to produce athree-channel full color image, that is, each pixel contains three colorchannel values, missing information needs to be estimated fromsurrounding pixels of CFA pattern raw data. This is called demosaicingalgorithm or process. The simplest demosaicing algorithm is linearinterpolation applied to every color channel. More advanced demosaicingmethods are summarized in X. Li, B. Gunturk, and L. Zhang, “Imagedemosaicing: a systematic survey,” Proc. SPIE, Vol. 6822, 68221J, 2008;D. Alleysson and B. C. de Lavarene, “Frequency selection demosaicking: areview and a look ahead,” Proc. SPIE-IS&T Electronic Imaging, Vol. 6822,68221M, 2008.

Sampling by color filter array (CFA) causes severe aliasing in additionto the aliasing caused by undersampling of many low-cost sensors, e.g.,the CCD aperture.

The resolution of a color image reconstructed from the demosaicingmethod is equal to the physical resolution of a CCD. However, theresolution of an image sensor can be improved by a digital imageprocessing algorithm: super-resolution image reconstruction.

Super-resolution image reconstruction generally increases imageresolution without necessitating a change in the design of the opticsand/or detectors by using a sequence (or a few snapshots) oflow-resolution images. Super-resolution image reconstruction algorithmseffectively de-alias undersampled images to obtain a substantiallyalias-free or, as identified in the literature, a super-resolved image.

When undersampled images have sub-pixel shifts between successiveframes, they contain different information regarding the same scene.Super-resolution image reconstruction involves, inter alia, combininginformation contained in undersampled images to obtain an alias-free(high-resolution) image. Super-resolution image reconstruction frommultiple snapshots, taken by a detector which has shifted in position,provides far more detail information than any interpolated image from asingle snapshot.

Methods for super-resolving images that are acquired from CFA sensorsare in the following references.

One of these methods is to capture multiple appropriately positioned CFAimages that are used to fill in the “holes” in a Bayer pattern, as forexample in U.S. Pat. No. 7,218,751, “Generating super resolution digitalimages,” May 15, 2007, A. M. Reed and B. T. Hannigan. Values of thepixels in multiple images which are appropriately aligned to each pixelposition are averaged to generate a better value for each pixelposition. In this method, in order to produce a useful result,information carried by a digital watermark is used to determine thealignment of the images.

Another type of these methods is called separate approach or two-passalgorithm. The CFA images are first demosaicked and then followed by theapplication of super-resolution, as for example in U.S. Pat. No.7,260,277, “Method for obtaining a high-resolution digital image,” Aug.21, 2007, G. Messina, S. Battiato, and M. Mancuso. Each CFA input imageis subjected to an interpolation phase to generate a completelow-resolution with three color channels containing three primary colorsin RGB format and is thus linearly transformed into a completelow-resolution image in the YCrCb format, where Y represents theluminance component, Cr and Cb represent two chrominance components.After this, a modified back projection super-resolution approach, basedin M. Irani and S. Peleg, “Super resolution from image sequence,”Proceedings of the 10th International Conference on Pattern Recognition,Vol. 2, pages 115-120, is applied only to the luminance component Y ofthe multiple images.

The third type of these methods is called joint or one-pass method inwhich demosaicing and super-resolution algorithms are simultaneouslycarried for a CFA image sequence, as for example in U.S. Pat. No.7,515,747, “Method for creating high resolution color image, system forcreating high resolution color image and program creating highresolution color image,” Apr. 7, 2009, M. Okutomi and T. Goto; U.S. Pat.No. 7,379,612, “Dynamic reconstruction of high-resolution video fromcolor-filtered low-resolution video-to-video super-resolution,” May 27,2008, P. Milanfar, S. Farsiu, and M. Elad. In this method, anobservation model is formulated to relate the original high-resolutionimage to the observed low-resolution images to incorporate colormeasurements encountered in video sequences. Then, the high-resolutionfull color image is obtained by solving an inverse problem by minimizinga cost function which is the function of the difference between theestimated low-resolution input images and the measured input images byimposing penalty terms, such as smoothness of chrominance andinter-color dependencies.

There is a need to produce high-resolution three channel full colorimages from a sequence of low-resolution images captured by a low-costCFA imaging device that has experienced translations and/or rotations.The amount of sub-pixel translation and/or rotation may be unknown andit creates a need to estimate sub-pixel translation and rotation. Inorder to produce high-resolution images from a sequence of CFAundersampled (low-resolution) images, there exists a need to eliminatealiasing, while taking into account natural jitter.

SUMMARY OF THE INVENTION

The present invention provides a method and system for thesuper-resolving images reconstructed from sequences of CFA lowerresolution imagery affected by undersampled color filter array sensor.

An embodiment of the present invention utilizes a sequence ofundersampled CFA low-resolution images from the same scene withsub-pixel translations and/or rotations among the images to reconstructan alias-free high-resolution three-channel full color image, or toproduce a sequence of such alias-free high-resolution full color framesas in a video sequence. The undersampled CFA low resolution images maybe captured in the presence of natural jitter or some kind of controlledmotion of the camera. For example, the natural jitter may be the resultof a vehicle traveling down a bumpy road or an aircraft experiencingturbulence. The present invention is adaptable to situations where priorknowledge about the high-resolution full color image is unknown, as wellwhen unknown, irregular or uncontrolled sub-pixel translations and/orrotations have occurred among the low-resolution CFA images.

As depicted in FIG. 7, a sequence of original input CFA low resolutionimages is passed into a transformation parameter estimation algorithm(Box 200). The transformation parameters include translation androtation values. The Box 210 includes three steps obtaining gross andsub-pixel transformation parameter estimation for the green channel,from which the demosaiced green channel sequence is obtained (Box 212),an estimate of the overall shift of each frame with respect to areference frame in the green channel sequence is obtained (Box 214), anda sub-pixel translation and/or rotation estimation is obtained (Box216). Since the luminance component in a full color image contains mostinformation and a Bayer color filter array is used in this embodiment,the green channel is chosen as the luminance component fortransformation parameter estimation. Because it is important to keep themosaicing order of the input CFA images, the input images are notaligned with respect to the gross-shift estimates. Rather, bothestimated gross and sub-pixel transformation parameters are assigned tothe red and blue channel sequences (Box 240 and 250). An error reductionalgorithm is applied (Box 260) to the input CFA low resolution imageswith the estimated gross translations and sub-pixel shifts and/orrotations among images to obtain the high-resolution (alias-free) fullcolor output.

According to a preferred embodiment of the present invention, there isprovided a system for improving picture quality of color images bycombing the content of a plurality of frames of the same subject;comprising:

at least one processor; the at least one processor comprising a memoryfor storing a plurality of frames of a subject; the at least oneprocessor operating to combine the content of plurality of frames of thesubject into a combined color image by performing:

a process in which at least two multicolored frames are converted tomonochromatic predetermined color frames;

a gross shift process in which the gross shift translation of onemonochromatic predetermined color frame is determined relative to areference monochromatic predetermined color frame;

a subpixel shift process utilizing a correlation method to determine thetranslational and/or rotational differences of one monochromaticpredetermined color frame to the reference monochromatic predeterminedcolor frame to estimate sub-pixel shifts and/or rotations between theframes; and

an error reduction process to determine whether the resolution of theresulting combined color image is of sufficient resolution; the errorreduction process comprising applying at least one spatial frequencydomain constraint and at least one spatial domain constraint to thecombined color image to produce at least one high-resolution full colorimage.

FIG. 8 shows an alternative embodiment of this invention. Since allluminance and chrominance channel images contain information from thescene, the transformation parameter estimation is obtained from allthree demosaiced color channels. The input CFA images and the estimatedgross shifts and sub-pixel shifts and/or rotations of green (Box 310 and320), red (Box 330 and 340), and blue (Box 350 and 360) channels areinput to the error-energy reduction algorithm (Box 370) to generate thehigh-resolution full color output image.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can best be understood when reading the followingspecification with reference to the accompanying drawings, which areincorporated in and form a part of the specification, illustratealternate embodiments of the present invention, and together with thedescription, serve to explain the principles of the invention. In thedrawings:

FIG. 1 is an illustration of the Bayer pattern in which for each 2×2 setof pixels, two diagonally opposed pixels have green filters, and theother two pixels have red and blue filters, which is called quincunxsampling pattern.

FIG. 2 illustrates a graphic view showing the relationship between twocoordinate planes, (u,v) with the origin at o_(uv) and (x,y) with theorigin at o_(xy); θ is the rotation angle, (x_(s),y_(s)) the relativeshifts in (x,y) coordinates when u=0 and v=0, (u_(s),v_(s)) the relativeshifts in (u,v) coordinates when x=0 and y=0.

FIG. 3 is a schematic representation of a generalized sequence ofcomputations and/or stages of the computational efficient translationestimation algorithm for two-dimensional images.

FIG. 4 is a schematic representation of a generalized sequence ofcomputations and/or stages of the computational efficient sub-pixeltranslation estimation algorithm for two-dimensional images.

FIG. 5 illustrates a graphic view showing the relationship between twocoordinate planes, (k_(u),k_(v)) and (k_(x),k_(y)) in the spatialfrequency domain in which the corresponding two origins coincide at o.

FIG. 6 is a schematic diagram of a generalized sequence of stepsrepresenting an algorithm for estimating translation and rotationtransformation parameters.

FIG. 7 is an overall schematic representation of an embodiment of thepresent invention for reconstructing super-resolved full color imagesfrom CFA low-resolution sequences whose green channel images are used toestimate gross translations and sub-pixel translations and rotationsamong frames.

FIG. 8 is an overall schematic representation of alternative embodimentof the present invention for reconstructing super-resolved full colorimages from CFA low-resolution sequences whose green, red, and bluechannel images are used to estimate gross translations and sub-pixeltranslations and rotations among frames.

FIG. 9 illustrates a graphic view showing the rectangular sampling grid.

FIG. 10 illustrates a graphic view showing the rectangular sampledspectrum.

FIG. 11 illustrates a graphic view showing the hexagonal sampling grid.

FIG. 12 illustrates a graphic view showing the hexagonal sampledspectrum.

FIG. 13 is a schematic diagram of sequence of steps representing analgorithm for reconstructing CFA green channel.

FIG. 14 shows the original gray scale image of a nature scene.

FIG. 15 shows the spectrum of the original image.

FIG. 16 shows the subsampled rectangular grid for red or blue imagewhich is subsampled by 2 from the original image.

FIG. 17 shows the red or blue sampled image.

FIG. 18 shows the spectrum of the red or blue sampled image, in whichthe repeated versions of the spectrum of the original image arepresented in the rectangular sampling fashion.

FIG. 19 shows the reconstructed red or blue sampled image.

FIG. 20 shows the hexagonal sampling grid for green image.

FIG. 21 shows the hexagonal sampled green image.

FIG. 22 shows the spectrum of the hexagonal sampled green image, inwhich the repeated versions of the spectrum of the original image arepresented in the hexagonal sampling fashion.

FIG. 23 shows the reconstructed green sampled image.

FIG. 24 is a graphic view showing an image acquisition model;

FIG. 25 is a graphic view showing factors that dictate the bandwidth ofthe measured target signature;

FIG. 26 is a graphic view depicting the bandwidth phenomenon in thesuper-resolution image reconstruction algorithm.

FIG. 27 is a schematic representation of a sequence of stepsrepresenting, inter alia, an error-energy reduction algorithm for CFAinput image sequences influenced or affected by both translations androtations.

FIG. 28 illustrates an example of super-resolving a full color imagefrom a sequence of CFA input images for a license plate scene. In orderto protect the license plate owner's privacy, part of license plate iscovered.

FIG. 29 illustrates an example of super-resolving a full color imagefrom a sequence of CFA input images for a car emblem scene. The letter“Ford” shows clearly in the super-resolved luminance image.

FIG. 30 depicts a high level block diagram of a general purposecomputer.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which embodiments of theinvention are shown. However, this invention should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart. In the drawings, the thickness of layers and regions may beexaggerated for clarity. Like numbers refer to like elements throughout.As used herein the term “and/or” includes any and all combinations ofone or more of the associated listed items.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to limit the full scope of theinvention. As used herein, the singular forms “a”, “an” and “the” areintended to include the plural forms as well, unless the context clearlyindicates otherwise. It will be further understood that the terms“comprises” and/or “comprising,” when used in this specification,specify the presence of stated features, integers, steps, operations,elements, and/or components, but do not preclude the presence oraddition of one or more other features, integers, steps, operations,elements, components, and/or groups thereof.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

With reference to FIG. 7, components of a preferred embodiment of thepresent invention include gross translation estimation (Box 214) andsub-pixel transformation estimation (Box 216). Gross translation (orshift) estimates are intended to obtain the overall shift of each imagewith respect to a reference image.

Sub-pixel transformation, also known as fraction pixel displacement,subpixel shift, or subpixel displacement, is estimated by a variety ofmethods in both patents and literature. Frame-to-frame motion detectionbased on gradient decent methods are most commonly used (see: (1) U.S.Pat. No. 5,767,987, “Method and apparatus for combining multiple imagescans for enhanced resolution,” Jun. 16, 1998, G. J. Wolff and R. J. VanSteenkiste; (2) U.S. Pat. No. 6,650,704, “Method of producing a highquality, high resolution image from a sequence of low quality, lowresolution images that are undersampled and subject to jitter,” Nov. 18,2003, R. S. Carlson, J. L. Arnold, and V. G. Feldmus; (3) U.S. Pat. No.6,285,804, “Resolution improvement from multiple images of a scenecontaining motion at fractional pixel values,” Sep. 4, 2001, R. J.Crinon and M. I. Sezan; (4) U.S. Pat. No. 6,349,154, “Method andarrangement for creating a high-resolution still picture,” Feb. 19,2002, R. P. Kleihorst). One of the variations of these methods is toestimate the velocity vector field measurement based on spatio-temporalimage derivative (see U.S. Pat. No. 6,023,535, “Methods and systems forreproducing a high resolution image from sample data,” Feb. 8, 2000, S.Aoki). Most of these methods need to calculate matrix inversion or useiterative methods to calculate the motion vectors. Bergen (see U.S. Pat.No. 6,208,765, “method and apparatus for improving image resolution,”Mar. 27, 2001, J. R. Bergen) teaches a method to use warp information toobtain sub-pixel displacement. Stone et al (see U.S. Pat. No. 6,628,845,“Method for subpixel registration of images,” Sep. 30, 2003, H. S.Stone, M. T. Orchard, E-C Chang, and S. Martucci) teaches another methodthat is to estimate the phase difference between two images to obtainsub-pixel shifts. In this method, the minimum least square solution hasto be obtained to find the linear Fourier phase relationship between twoimages. U.S. Pat. No. 7,602,997, invented by the inventor herein,utilizes a correlation method without solving the minimum least squareproblem to explore the translational differences (shifts in x and ydomains) of Fourier transforms of two images to estimate sub-pixelshifts between two low resolution aliased images.

Translation and Rotation Estimation

Two image frames of a low-resolution image sequence may be designated ƒ₁(x,y) and ƒ₂(u,v). Because of the conditions under which the frames areacquired, the second image is a shifted and/or rotated version of thefirst image, that is,

$\begin{matrix}{{f_{2}\left( {u,v} \right)} = {f_{1}\left( {x,y} \right)}} & {{Equation}\mspace{14mu}\left( {1A} \right)} \\{\begin{bmatrix}u \\v\end{bmatrix} = {\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}{x - x_{s}} \\{y - y_{s}}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {2A} \right)} \\{or} & \; \\{\begin{bmatrix}u \\v\end{bmatrix} = {{\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}} + \begin{bmatrix}u_{s} \\v_{s}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {3A} \right)}\end{matrix}$where (x,y) represent the coordinate plane in the first image, and (u,v)represent the coordinate plane in the shifted and rotated second image,as shown in FIG. 2; θ is the rotation angle and (x_(s),y_(s)) therelative shifts between ƒ₁(x,y) and ƒ₂(u,v) in (x,y) plane (where u=0and v=0) and u_(s),v_(s)) the relative shifts in (u,v) plane (where x=0and y=0) (as shown in FIG. 2). The relation between (x_(s),y_(s)) and(u_(s),v_(s)) can be written as:

$\begin{matrix}{\begin{bmatrix}u_{s} \\v_{s}\end{bmatrix} = {\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}{- x_{s}} \\{- y_{s}}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {4A} \right)}\end{matrix}$If the relative shifts (u_(s),v_(s)) are obtained, the shifts(x_(s),y_(s)) can be obtained. Hereinafter, both (x_(s),y_(s)) and(u_(s),v_(s)) may be used for the relative shifts. The transformationparameters between images including translation (shift) (x_(s),y_(s))and rotation θ are estimated using a signal registration method andproperties of Fourier transform.

Signal Registration—The signal registration method is described in thefollowing. Almost all of the transformation parameter estimation methodsare based on the signal registration method. It is assumed that thesignals to be matched differ only by an unknown translation. Inpractice, beside translation, there may be rotation, noise, andgeometrical distortions. Nevertheless, the signal registrationdescription offers a fundamental solution to the problem. The procedurefor pixel-level registration is to first calculate the correlationbetween two images ƒ₁(x,y) and ƒ₂(x,y) as:

$\begin{matrix}{{r\left( {k,l} \right)} = {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{R\left\lbrack {{f_{1}\left( {{n + k - 1},{m + l - 1}} \right)},{f_{2}\left( {n,m} \right)}} \right\rbrack}}}} & {{Equation}\mspace{14mu}\left( {5A} \right)}\end{matrix}$where 1≦k≦N, 1≦l≦M and R is a similarity measure, such as a normalizedcross-correlation or an absolute difference function. For the (N−1, M−1)samples of r(k,l), there is a maximum value of r(x_(s),y_(s)), where(x_(s),y_(s)) is the displacement in sampling intervals of ƒ₂(x,y) withrespect to ƒ₁(x,y). Many transformation parameter estimation methods arebased on the variations of the signal registration.

A signal registration based correlation method is used for translationparameter estimation between two images based on 1) the correlationtheorem; and 2) the shift property of Fourier transform. This methodbegins with the cross-correlation between two images. The terminology“cross-correlation” or “cross correlation” as used herein is a measureof similarity of two frames in terms of a function of one of them. Thetwo-dimensional normalized cross-correlation function measures thesimilarity for each translation:

$\begin{matrix}{{C\left( {k,l} \right)} = \frac{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{{f_{1}\left( {n,m} \right)}{f_{2}\left( {{n + k - 1},{m + l - 1}} \right)}}}}{\sqrt{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{f_{2}^{2}\left( {{n + k - 1},{m + l - 1}} \right)}}}}} & {{Equation}\mspace{14mu}\left( {6A} \right)}\end{matrix}$where 1≦k≦N, 1≦l≦M and (N,M) is the dimension of the image. If ƒ₁(x,y)matches ƒ₂(x,y) exactly, except for an intensity scale factor, at atranslation of (x_(s),y_(s)), the cross-correlation has its peak atC(x_(s),y_(s)). Various implementations and discussions can be found forexample in A. Rosenfeld and A. C. Kak, Digital Picture Processing, Vol.II pp. 36-39, Orlando, Fla.: Academic Press, 1982; L. G. Brown, “Asurvey of image registration techniques,” ACM Computing Survey, Vol. 24,No. 4, December 1992, pp. 325-376), both of which are herebyincorporated by reference.

Correlation Theorem More efficient signal registration method fortranslation estimation is using the correlation method in the spatialfrequency domain via the Correlation Theorem. The Correlation Theoremstates that the Fourier transform of the cross-correlation of two imagesis the product of the Fourier transform of one image and the complexconjugate of the Fourier transform of the other, that is,Q ₁₂(k _(x) ,k _(y))=ℑ[C(x,y)]=F ₂(k _(x) ,k _(y))·F ₁*(k _(x) ,k_(y))  Equation (7A)where F₁(k_(x),k_(y)) and F₂(k_(x),k_(y)) are Fourier transforms of twoimages, ƒ₁(x,y) and ƒ₂(x,y), respectively, F₁*(k_(x),k_(y)) is theconjugate of F₁(k_(x),k_(y)).

Fourier Transform Properties—The properties of Fourier transform aredescribed in the following that include shift property of Fouriertransform, polar transformation and polar function mapping, and rotationproperty of Fourier transform. A Fourier transform is an operation thattransforms one complex-valued function of a real variable into a newfunction. In such applications as image processing, the domain of theoriginal function is typically space and is accordingly called thespatial domain. The domain of the new function is spatial frequency, andso the Fourier transform is often called the spatial frequency domain(or Fourier domain) representation of the original function as itdescribes which frequencies are present in the original function. Asused herein the terminology “Fourier transform” refers both to thefrequency domain representation of a function and to the process orformula that “transforms” one function into the other.

Shift Property of Fourier Transform—Two members of the image sequenceare denoted by ƒ₁(x,y) and ƒ₂(x,y). The second image is a shiftedversion of the first image, such as,ƒ₂(x,y)=ƒ₁(x−x _(s) ,y−y _(s))  Equation (8A)where (x_(s),y_(s)) are the relative shifts (translational differences)between ƒ₁(x,y) and ƒ₂(x,y). Then,F ₂(k _(x) ,k _(y))=F ₁(k _(x) ,k _(y))exp[−j(k _(x) ,x _(s) +k _(y) y_(s))]  Equation (9A)where F₁(k_(x),k_(y)) and F₂(k_(x),k_(y)) are Fourier transforms of twoimages. Thus, a shift in one domain results in the addition of a linearphase function in the Fourier domain.

Polar Transform and Polar Function Mapping—The polar transformation ofthe domain (x,y) for a 2-D signal ƒ(x,y) may be represented by

$\begin{matrix}{{f_{p}\left( {\theta,r} \right)} \equiv {f\left( {x,y} \right)}} & {{Equation}\mspace{14mu}\left( {10A} \right)} \\{where} & \; \\{\theta \equiv {\arctan\left( \frac{y}{x} \right)}} & {{Equation}\mspace{14mu}\left( {11{AA}} \right)} \\{and} & \; \\{r \equiv \sqrt{x^{2} + y^{2}}} & {{Equation}\mspace{14mu}\left( {11{AB}} \right)}\end{matrix}$Note that ƒ(x,y)≠ƒ_(p)(x,y). This equation is called the polar functionmapping of ƒ(x,y). The polar function mapping of F(k_(x),k_(y)) in thespatial frequency domain may be defined via

$\begin{matrix}{{F_{p}\left( {\phi,\rho} \right)} \equiv {F\left( {k_{x},k_{y}} \right)}} & {{Equation}\mspace{14mu}\left( {12A} \right)} \\{where} & \; \\{\phi \equiv {\arctan\left( \frac{k_{y}}{k_{x}} \right)}} & {{Equation}\mspace{14mu}\left( {13{AA}} \right)} \\{and} & \; \\{\rho \equiv \sqrt{k_{x}^{2} + k_{y}^{2}}} & {{Equation}\mspace{14mu}\left( {13{AB}} \right)}\end{matrix}$It is noted that F_(p)(•,•) is not the Fourier transform of ƒ_(p)(•,•).

Rotation Property of Fourier Transform—Two members of the image sequenceare denoted ƒ₁(x,y) and ƒ₂(u,v), the second image being a rotatedversion of the first image,

$\begin{matrix}{{f_{2}\left( {u,v} \right)} = {f_{1}\left( {x,y} \right)}} & {{Equation}\mspace{14mu}\left( {1A} \right)} \\{\begin{bmatrix}u \\v\end{bmatrix} = {\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {3A} \right)}\end{matrix}$where θ is the relative rotation (rotational differences) betweenƒ₁(x,y) and ƒ₂(u,v). Then,F ₂(k _(u) ,k _(v))=F ₁(k _(x) ,k _(y))  Equation (14A)where [k_(u), k_(v)] and [k_(x),k_(y)] have the following relation,

$\begin{matrix}{\begin{bmatrix}k_{u} \\k_{v}\end{bmatrix} = {\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}k_{x} \\k_{y}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {15A} \right)}\end{matrix}$Using the polar function mappings of F₁(k_(y),k_(y)) andF₂(k_(u),k_(v)), thenF _(p2)(φ,ρ)=F _(p1)(φ−θ,ρ)  (Equation 16A)That is, if one image is a rotated version of another image, then thepolar function mapping of one is the shifted version of the polarfunction mapping of the other one in the spatial frequency domain.

Gross Translation Estimation Theory

The gross translation estimation (Box 214, and, alternatively, Boxes320, 340, 360) is described in the following. The second image can beexpressed as a shifted version of the first image, such as,ƒ₂(u,v)=ƒ₁(x,y) Equation (1A)

$\begin{matrix}{\begin{bmatrix}u \\v\end{bmatrix} = \begin{bmatrix}{x - x_{s}} \\{y - y_{s}}\end{bmatrix}} & {{Equation}\mspace{14mu}\left( {17A} \right)}\end{matrix}$where (x_(s),y_(s)) are the relative shifts (translational differences)between ƒ₁(x,y) and ƒ₂(u,v). The shifts (x_(s),y_(s)) are foundaccording to the Correlation Theorem based on the signal registrationmethod by exploring the translational differences (shifts in x and ydomains) of Fourier transforms of two images.

According to the Correlation Theorem, the Fourier transform of thecross-correlation of the two images, ƒ₁(x,y) and ƒ₂(x,y) can be found asfollows:Q ₁₂(k _(x) ,k _(y))=ℑ[C(X,y)]=F ₂(k _(x) ,k _(y))·F ₁*(k _(x) ,k_(y))  Equation (7A)where F₁(k_(x),k_(y)) and F₂(k_(x),k_(y)) are Fourier transforms of twoimages, ƒ₁(x,y) and ƒ₂(x,y), respectively, F₁*(k_(x),k_(y)) is theconjugate of F₁(k_(y),k_(y)).

Using the shift property of Fourier transform, the Fourier transform,F₂(k_(x),k_(y)), can be substituted viaF ₂(k _(x) ,k _(y))=F ₁(k _(x) ,k _(y))exp[−j(k _(x) x _(s) +k _(y) y_(s))]  Equation (9A)Then, the Fourier transform of the cross-correlation of the two images,ƒ₁(x,y) and ƒ₂(x,y), that is, the product of the Fourier transform ofone image and the complex conjugate of the Fourier transform of theother, can then be expressed as

$\begin{matrix}\begin{matrix}{{Q_{12}\left( {k_{x},k_{y}} \right)} = {{F_{1}\left( {k_{x},k_{y}} \right)}{F_{1}^{*}\left( {k_{x},k_{y}} \right)}{\exp\left\lbrack {- {j\left( {{k_{x}x_{s}} + {k_{y}y_{s}}} \right)}} \right\rbrack}}} \\{= {{{F_{1}\left( {k_{x},k_{y}} \right)}}{\exp\left\lbrack {- {j\left( {{k_{x}x_{s}} + {k_{y}y_{s}}} \right)}} \right\rbrack}}}\end{matrix} & {{Equation}\mspace{14mu}\left( {18A} \right)}\end{matrix}$Then, the shifts (x_(s), y_(s)) are found as the peak of inverse Fouriertransform of the cross-correlation, Q₁₂(k_(x),k_(y)).

Translation Estimation with Aliasing Consideration—Since both imagesƒ₁(x,y) and ƒ₂(u,v) are aliased, i.e., the high frequency components arefolded into a portion of low frequency components. In general, for analiased image, the image energy at lower frequency components isstronger than the folded high frequency components. Thus, the signal toaliasing power ratio is relatively high around (k_(x),k_(y))≈(0,0).Therefore, a low-pass filtering, W, is applied to the Fourier transformsF₁(k_(x),k_(y)) and F₂(k_(x),k_(y)). A schematic diagram of thecomputational efficient translation estimation method as described inU.S. Pat. No. 7,602,997, “Method of Super-Resolving Images,” filed Jan.19, 2005, S. S. Young, hereby incorporated by reference, is shown inFIG. 3.

Since the Fourier transform can be computed efficiently with existing,well-tested programs of FFT (fast Fourier transform) (and occasionallyin hardware using specialized optics), the use of the FFT becomes mostbeneficial for cases where the images to be tested are large asdescribed in L. G. Brown, “A survey of image registration techniques,”ACM Computing Survey, Vol. 24, No. 4, December 1992, pp. 325-376.

Sub-Pixel Translation Estimation

Translation Estimation with Sub-pixel Accuracy and AliasingConsideration—In super-resolution image reconstruction of a preferredembodiment, sub-pixel shifts (or fractional pixel displacements) amongthe input images are estimated using a method to estimate or calculateframe-to-frame motion within sub-pixel accuracy. In general, thesmallest motion that can be obtained from conventional motion estimationis one pixel. Obtaining sub-pixel motion accurately is notstraightforward. A method to achieve the sub-pixel accuracy solution hasbeen described in the aforementioned U.S. Pat. No. 7,602,997. Thismethod is based on resampling via frequency domain interpolation byusing a Fourier windowing method as described in S. S. Young,“Alias-free image subsampling using Fourier-based windowing methods,”Optical Engineering, Vol. 43, No. 4, April 2004, pp. 843-855 (herebyincorporated by reference), called power window.

Because the low-resolution input images contain aliasing, a low-passfilter is applied to the input images first. A schematic diagram of thecomputational efficient sub-pixel translation estimation method asdescribed in U.S. Pat. No. 7,602,997 is shown in FIG. 4.

Theory of Translation and Rotation Estimation

The sub-pixel translation and rotational estimation is described in thefollowing. Assuming the second image is a shifted and rotated version ofthe first image, i.e.,

$\begin{matrix}{{f_{2}\left( {u,v} \right)} = {f_{1}\left( {x,y} \right)}} & {{Equation}\mspace{14mu}\left( {1A} \right)} \\{\begin{bmatrix}u \\v\end{bmatrix} = {{\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}} + \begin{bmatrix}u_{s} \\v_{s}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {3A} \right)}\end{matrix}$where θ is the relative rotation and (u_(s),v_(s)) the relative shiftsbetween ƒ₁(x,y) and ƒ₂(u,v).

From the shift property of Fourier transform of these images, thefollowing relation is derived:F ₂(k _(u) ,k _(v))=F ₁(k _(x) ,k _(y))·exp[−j(k _(u) u _(s) +k _(v) v_(s))]  Equation (9A)where F₂(k_(u),k_(v)) and F₁(k_(x),k_(y)) are the Fourier transforms ofthe two images, and the (k_(u),k_(v)) domain is the rotated version ofthe (k_(x),k_(y)) domain; that is,

$\begin{matrix}{\begin{bmatrix}k_{u} \\k_{v}\end{bmatrix} = {\begin{bmatrix}{\cos\;\theta} & {\sin\;\theta} \\{{- \sin}\;\theta} & {\cos\;\theta}\end{bmatrix}\begin{bmatrix}k_{s} \\k_{y}\end{bmatrix}}} & {{Equation}\mspace{14mu}\left( {15A} \right)}\end{matrix}$FIG. 5 shows the transformation of the two coordinates, (k_(u),k_(v))and (k_(x),k_(y)), in the spatial frequency domain in which thecorresponding two origins coincide.

Then from the polar transformation, the rotation and shift properties ofFourier transform as described previously, the polar function mappings(see operation in Equation 12A) of the 2-D Fourier transforms of the twoimages are related byF _(p2)(φ,ρ)=F _(p1)(φ−θ,ρ)·exp[−j(k _(u) u _(s) +k _(v) v_(s))]  Equation (19A)

Thus, for their magnitudes we have,|F _(p2)(φ,ρ)|=|F _(p1)(φ−θ,ρ)  Equation (20A)That is, if one image is a rotated and shifted version of the otherimage, then the magnitude of polar function mapping of one is theshifted version of the magnitude of polar function mapping of the otherone in the spatial frequency domain.

Note the difference between Equation (20A) and Equation (16A). One imageis represented as a rotated version of the other image in Equation(16A), while one image is represented as a rotated and shifted versionof the other image in Equation (20A).

Therefore, the rotation angle difference θ is represented as atranslational difference using the polar transformation and polarfunction mapping of Fourier transform. Using the correlation method, therotation angle between the two images is estimated in a manner similarto the translation estimation.

The rotation estimation is described in the following in detail. Becausethe rotation difference is only represented in the first domain of theFourier transform, the rotation angle θ is found as the peak of inverseof the marginal 1-D Fourier transform of the cross-correlation withrespect to φ. The marginal 1-D Fourier transform is an operation ofperforming the one-dimensional Fourier transform of a 2-D functionƒ(x,y) with respect to one of the variables, x or y.

The marginal 1-D Fourier transform of the cross-correlation with respectto φ between the two images is defined as:Q _(P12)(k _(φ),ρ)=ℑ_(p2)(k _(φ),ρ)ℑ_(p1)*(k _(φ),ρ)  Equation (21A)where ℑ_(p1)(k_(φ),ρ) and ℑ_(p2)(k_(φ),ρ) are the marginal 1-D Fouriertransforms with respect to φ for the two images |F_(p1)(φ,ρ)| and|F_(p2)(φ,ρ)| (as defined in Equation 12A), respectively, and ℑ_(p1)⁺(k_(φ),ρ) is the complex conjugate of ℑ_(p1)(k_(φ),ρ).

After the rotation angle is estimated, the second image is rotated backby the angle −θ to form the backward rotated image ƒ_(2θ)(x,y). Then theshifts (x_(s),y_(s)) between the two images are estimated betweenƒ_(2θ)(x,y) and ƒ₁(x,y) according to the correlation method based on theCorrelation Theorem and the shift property of Fourier transform asdiscussed previously.

Sub-Pixel Rotation and Translation Estimation

Rotation and Translation Estimation with Sub-pixel Accuracy—Similar asin the translation estimation with sub-pixel accuracy, the rotationestimation with sub-pixel accuracy is achieved by resampling viafrequency domain interpolation. However, the resampling operation isonly performed in the φ domain (as defined in Equation 13A) since therotation angle is the only parameter to be estimated. The subsequentsub-pixel translation estimation, (x_(s),y_(s)) between the backwardrotated image ƒ_(2θ)(x,y) and the first image ƒ₁(x,y), is achieved byresampling for both k_(x) and k_(y) domains via frequency domaininterpolation (see FIG. 5).

Rotation with Aliasing Consideration The common center area of the twoimages ƒ₁(x,y) and ƒ₂(u,v) contributes most to the rotation angleestimation. A radially symmetrical window is applied to both images toemphasize the center common region between the two images. This radiallysymmetrical window is also designed as a low-pass filter to achieveaccurate rotation estimation by filtering out the aliased high frequencycomponents in the input images. As mentioned previously, the terminologyaliased refers to the high frequency components folded into a portion oflow frequency components. In general, for an aliased image, the imageenergy at lower frequency components is stronger than the folded highfrequency components. Thus, the signal to aliasing power ratio isrelatively high around (k_(x),k_(y))≈(0,0).

The radially symmetrical and low-pass window, W, is applied to theFourier transforms F₁(k_(x),k_(y)) and F₂(k_(u),k_(v)) in the first stepof the rotation estimation procedure, that is,{tilde over (F)} ₁(k _(x) ,k _(y))=F ₁(k _(x) ,k _(y))·W(k _(x) ,k_(y))  Equation (22AA){tilde over (F)} ₂(k _(u) ,k _(v))=F ₂(k _(u) ,k _(v))·W(k _(u) ,k_(v))  Equation (22AB)Sub-Pixel Rotation and Translation Estimation Procedure

A procedure to estimate translation and rotation parameters withsub-pixel accuracy between two images, ƒ₁(x,y) and ƒ₂(u,v), issummarized in the following. FIG. 6 shows a schematic representation ofthis estimating algorithm, which may comprise:

-   1) Input two low-resolution images ƒ₁(x,y) and ƒ₂(u,v) (Box 30).-   2) Obtain the 2-D Fourier transforms of the two images,    F₁(k_(x),k_(y)) and F₂(k_(u), k_(v)) (Box 31).-   3) Apply a radially symmetrical window W to the Fourier transforms    of the two images to obtain {tilde over (F)}₁(k_(x),k_(y)) and    {tilde over (F)}₂(k_(u),k_(v)) (Box 32), see Equation (22AA)-(22AB).-   4) Construct the magnitudes of the polar function mappings of the    two windowed images, |F_(P1)(φ,ρ)| and |F_(P2)(φ,ρ)| (Box 33), as    defined in Equation (12A).-   5) Obtain the marginal 1-D Fourier transforms with respect to φ for    the magnitude of the two polar function mappings, ℑ_(P1) (k_(φ), ρ)    and ℑ_(P2)(k_(φ),ρ) (Box 34).-   6) Form the cross-correlation function of the two images,    ℑ_(P1)(k_(φ),ρ) and ℑ_(P2)(k_(φ),ρ), to obtain Q_(p12)(k_(φ),ρ) (Box    35), see Equation (21A).-   7) Estimate the rotation angle θ between two images ƒ₁(x,y) and    ƒ₂(u,v) with sub-pixel accuracy from the cross-correlation function    Q_(p12)(k_(φ),ρ) (Box 36).-   8) Rotate the second image by the angle −θ to form the backward    rotated image ƒ_(2θ)(x,y) (Box 37).-   9) Estimate the relative shift between ƒ_(2θ)(x,y) and ƒ₁(x,y) with    sub-pixel accuracy to obtain the relative shifts (x_(s),y_(s)) (Box    38).    Example of Sub-Pixel Rotation and Translation

An illustration of translation and rotation estimation with sub-pixelaccuracy is shown in FIGS. 7a, 7b, 8, and 9 of application Ser. No.12/576,132. In the featured example, a high-resolution forward lookinginfrared (FLIR) image was used to simulate a sequence of low-resolutionundersampled images with translations and rotations.

An original FLIR tank image is shown in FIG. 7 a of application Ser. No.12/576,132. The size of this image is 40×76. First, we up-sample thisimage using the Fourier windowing method by a factor of 30 to obtain asimulated high-resolution image with a size of 1200×2280 as shown inFIG. 7 b of application Ser. No. 12/576,132. Then the low-resolutionimages are generated from this simulated high-resolution image bysub-sampling and rotating. The randomly selected translation androtation parameters for six of each low-resolution image are listed asfollows:

-   -   x_(s)=[0 3.1 7.7 12.3 14.1 17.3];    -   y_(s)=[0 2.3 8.5 11.2 15.2 16.8];    -   θ=[0 14.4 15.6 17.8 19.1 22.7];        The units of x_(s) and y_(s) are pixels and the unit of θ is        degrees. FIG. 8 of application Ser. No. 12/576,132 shows six        simulated low-resolution images with shifts and rotates among        them.

After applying the correlation method for sub-pixel shift and rotationestimation to these 6 simulated low-resolution images, the sub-pixelshift and rotation values are obtained. The estimated sub-pixel shiftand rotation values are all correct according to the actual values. FIG.9 of application Ser. No. 12/576,132 shows the transformation of theplane of one image (image 6) into the target domain of the referenceimage (image 1).

FIG. 7 is an overall schematic representation of a preferred embodimentof the present invention. A sequence of original input CFA lowresolution images is passed into a transformation (including translationand rotation) parameter estimation algorithm (Box 200) that includesobtaining gross and sub-pixel transformation parameter estimationalgorithm (Box 210) for the green channel, from which the reconstructedgreen channel sequence is obtained (Box 212), an estimate of the overallshift of each frame with respect to a reference frame in the greenchannel sequence is obtained (Box 214), and a sub-pixel translation androtation estimation algorithm (Box 216). Since the luminance componentin a full color image contains most information and a Bayer color filterarray is used in this embodiment, the green channel is chosen as theluminance component for transformation parameter estimation. Because itis important to keep the mosaicing order of the input CFA images, theinput images are not aligned with respect to the gross-shift estimates.Rather, both estimated gross and sub-pixel transformation parameters areassigned to the red and blue channel sequences (Box 240 and 250).

The method of acquiring a sequence of images in super-resolution imagereconstruction in this embodiment may involve non-controlled sub-pixeltranslations and rotations; e.g. natural jitter. This can be more costeffective and more practical than the controlled sub-pixel translationand rotation image acquisition. In many applications, the imager may becarried by a moving platform or the object itself may be moving. Inmilitary reconnaissance situations, the moving platform may be a person,an unmanned ground vehicle (UGV), or an unmanned aerial vehicle (UAV)flying in a circular motion or a forward motion. As a result, acquiredimages may contain sub-pixel translations and/or rotations. During theprocess of super-resolution image reconstruction, after multiplesub-pixel translated and/or rotated CFA images are obtained, they arethen combined to reconstruct the high-resolution full color image byprojecting low-resolution CFA images onto three high-resolution gridsrepresenting red, green, and blue channels, respectively. Each grid isof a finer sampled grid than the CFA lower resolution grid. Thesub-pixel translations and rotations found in images resulting fromnatural jitter pose a need for accurate estimation.

In Box 200 of FIG. 7, the gross translations and the sub-pixeltranslations and rotations are obtained for each color channel sequence.In Box 260 of FIG. 7, an error reduction algorithm is applied to theinput CFA low resolution images with the estimated gross shifts andsub-pixel shifts and/or rotations among images to obtain thehigh-resolution (alias-free) full color output. The output may be eithera single high-resolution full color image that is generated from asequence of CFA low-resolution images or a sequence of high-resolutionfull color images that are generated from multiple sequences of CFAlow-resolution images.

The procedure of obtaining the reconstructed green channel image (Box212) is first described in the following.

Fourier Domain Reconstructing from Incomplete Sampled Data Based onSpectral Properties of Rectangular and Hexagonal Sampling

Green Channel Candidate—One important step in super-resolution is toaccurately estimate sub-pixel shifts and rotations among low-resolutionimages. Since the undersampled CFA images are captured either by naturaljitter or some kind of controlled motion of the camera, the images ofsuccessive frames contain not only the sub-pixel shifts but also theinteger pixel shifts.

Because CFA images contain severe aliasing, using the direct raw inputimages does not give accurate shift estimation. Using low-pass filteredversion of the input images is a better choice because the low-frequencyspectrum is less corrupted by aliasing.

In general, it was found in K. A. Prabhu and A. N. Netravali, “Motioncompensated component color coding,” IEEE Trans. Communications, Vol.COM-30, pp. 2519-2527, December 1982; R. C. Tom and A. K. Katsaggelos,“Resolution enhancement of monochrome and color video using motioncompensation,” IEEE Trans. Image Processing, Vol. 10, No. 2, February2001; U.S. Pat. No. 7,515,747, “Method for creating high resolutioncolor image, system for creating high resolution color image and programcreating high resolution color image,” Apr. 7, 2009, M. Okutomi and T.Goto, that it was sufficient to estimate the motion field of a colorsequence based on the luminance component only. Since a Bayer colorfilter array is used in this embodiment, the green channel is chosen asthe luminance component for transformation parameter estimation.However, when the other CFAs are used, the luminance channel is chosenbased on that CFA to be used.

The green channel in the Bayer pattern is sampled using a hexagonalsampling pattern. Obtaining the reconstructed green signal is equivalentto the process of reconstructing the original green signal at each pixellocation from hexagonal sampled green signal in the Bayer pattern.Before describing the hexagonal sampling, we describe the properties ofa more common sampling pattern, rectangular sampling.

Two-Dimensional Rectangular Sampling—Let ƒ_(δ)(x,y) be a two-dimensionaldelta-sampled signal of an arbitrary signal, ƒ(x,y):

$\begin{matrix}{{f_{\delta}\left( {x,y} \right)} = {{f\left( {x,y} \right)}{\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}\;{\delta\left( {{x - {m\;\Delta_{x}}},{y - {n\;\Delta_{y}}}} \right)}}}}} \\{= {\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}{{f\left( {x_{m},y_{n}} \right)}{\delta\left( {{x - x_{m}},{y - y_{n}}} \right)}}}}}\end{matrix}$where x_(m)=mΔ_(x) and y_(n)=nΔ_(y), Δ_(x) and Δ_(y) are the samplespacings in the spatial x and y domain respectively. Then, the Fouriertransform of the delta-sampled signal, F_(δ)(k_(x),k_(y)), is composedof repeated versions of the Fourier transform of the continuous signalF(k_(x),k_(y)), that is,

${F_{\delta}\left( {k_{x},k_{y}} \right)} = {\frac{1}{\Delta_{x}\Delta_{y}}{\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}{F\left( {{k_{x} - \frac{2\pi\; m}{\Delta_{x}}},{k_{y} - \frac{2\pi\; n}{\Delta_{y}}}} \right)}}}}$The repeated positions of this Fourier transform are at

$\left( {\frac{2\pi\; m}{\Delta_{x}},\frac{2\pi\; n}{\Delta_{y}}} \right)$for m=(−∞,∞) and n=(−∞,∞). Thus, provided that F(k_(x),k_(y)) has afinite support in the spatial frequency domain, for example,

${{F\left( {k_{x},k_{y}} \right)} = {{{{{0\mspace{14mu}{for}\mspace{14mu}{k_{x}}} \geq \frac{k_{x\; 0}}{2}}\&}{k_{y}}} \geq \frac{k_{y\; 0}}{2}}},$where

${k_{x\; 0} = {{\frac{2\pi}{\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{y\; 0}} = \frac{2\pi}{\Delta_{y}}}},$then ƒ(x,y) can be recovered from ƒ_(δ)(x,y) via lowpass filter in thefrequency domain. FIG. 9 illustrates a 2-D rectangular sampling grid.FIG. 10 illustrates the Fourier transform of F_(δ)(k_(x),k_(y)), whichis the repeated version of the continuous signal F(k_(x),k_(y)). Thecutoff frequency in spatial frequency domain is

$k_{{xs}\; 0} = {{\frac{2\pi}{2\;\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{{ys}\; 0}} = \frac{2\pi}{2\;\Delta_{y}}}$as shown in FIG. 10. The cutoff frequency in radial spatial frequencydomain is

$\rho_{s\; 0} = {\frac{2\pi}{2}\rho_{0}}$where ρ(k_(x),k_(y))=√{square root over (k_(x) ²+k_(y) ²)}, ρ₀=√{squareroot over ((1/Δ_(x))²+(1/Δ_(y))²)}{square root over((1/Δ_(x))²+(1/Δ_(y))²)}.

Spectral Properties of CFA Blue and Red Channel with Two-DimensionalRectangular Sampling—For CFA blue or red channel, it is atwo-dimensional rectangular sampling. However, if the sampling spaces ofthe CFA pattern in x and y domains are Δ_(x) and Δ_(y), respectively,the sampling spaces of blue or red channel are Δ_(x,C)=2Δ_(x) andΔ_(y,C)=2Δ_(y), where Cis for red or blue. Therefore, the cutofffrequency for red or blue channel is

$k_{{{xs}\; 0},C} = {{\frac{2\pi}{4\;\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{{{ys}\; 0},C}} = \frac{2\pi}{4\;\Delta_{y}}}$in spatial frequency domain,

$\rho_{{s\; 0},C} = {\frac{2\pi}{4}\rho_{0}}$in radial frequency domain.

Two-Dimensional Hexagonal Sampling—As used herein the terminology“hexagonal sampling” is depicted in the insert in FIG. 11; and refers tothe pattern formed by replacing the red and blue pixels (in the Bayerpattern shown in FIG. 1) with zero values.

As discussed in M. Soumekh, Fourier Array Imaging, Prentice Hall, 1994,pp. 167-177, the hexagonal sampling has the following lineartransformation:

$\begin{bmatrix}\alpha \\\beta\end{bmatrix} = {\begin{bmatrix}{T_{1}\left( {x,y} \right)} \\{T_{2}\left( {x,y} \right)}\end{bmatrix} = \begin{bmatrix}{{ax} + {by}} \\{{cx} + {dy}}\end{bmatrix}}$Note that rectangular sampling corresponds to the case of b=c=0. Theinverse of the hexagonal transformation is also linear, that is,

$\begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}{{A\;\alpha} + {B\;\beta}} \\{{C\;\alpha} + {D\;\beta}}\end{bmatrix}$The Jacobian of the hexagonal transformation is

${\frac{\partial\left( {\alpha,\beta} \right)}{\partial\left( {x,y} \right)}} = {{{T^{\prime}\left( {x,y} \right)}} = {{{ad} - {bc}}}}$Next, defineg(α,β)=ƒ(x,y)or, equivalentlyg[T ₁(x,y),T ₂(x,y)]=ƒ(x,y)It is assumed that the imaging system provides evenly spaced samples ofg(α,β) in the (α,β) measurement domain, for example, α_(m)=mΔ_(α) andβ_(n)=nΔ_(β). Our task is to reconstruct the function ƒ(x,y) from theavailable samples at (x_(mn),y_(mn)), whereα_(m) =mΔ _(α) =T ₁(x _(mn) ,y _(mn))β_(n) =nΔ _(β) =T ₂(x _(mn) ,y _(mn))

It is shown following in M. Soumekh, Fourier Array Imaging, PrenticeHall, 1994, pp. 167-177,δ(x−x _(m) ,y−y _(n))=|T′(x,y)|δ(α−α_(m),β−β_(n)).The delta-sampled signal is defined via

$\begin{matrix}{{f_{\delta}\left( {x,y} \right)} = {{f\left( {x,y} \right)}{\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}\;{\delta\left( {{x - {m\;\Delta_{x}}},{y - {n\;\Delta_{y}}}} \right)}}}}} \\{= {\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}{{f\left( {x_{m},y_{n}} \right)}{\delta\left( {{x - x_{m}},{y - y_{n}}} \right)}}}}}\end{matrix}$Using above equation, ƒ_(δ)(x,y) is rewritten as follows:

$\left. {{f_{\delta}\left( {x,y} \right)} = {{f\left( {x,y} \right)}{{T^{\prime}\left( {x,y} \right)}}{\sum\limits_{m = {- \infty}}^{\infty}{\sum\limits_{n = {- \infty}}^{\infty}{\delta\left\lbrack {{{T_{1}\left( {x,y} \right)} - {m\;\Delta_{\alpha}}},{{T_{2}\left( {x,y} \right)} - {n\;\Delta_{\beta}}}} \right)}}}}} \right\rbrack$Then, the Fourier transform of the delta-sampled signal,F_(δ)(k_(x),k_(y)), is composed of repeated versions of the Fouriertransform of the continuous signal F(k_(x),k_(y)), that is,

${F_{\delta}\left( {k_{x},k_{y}} \right)} = {\frac{1}{\Delta_{x}\Delta_{y}}{\sum\limits_{m = {- \infty}}^{\infty}{\sum\limits_{n = {- \infty}}^{\infty}{F\left\lbrack {{k_{x} - {\frac{2\pi\; m}{\Delta_{\alpha}}{T_{1}\left( {x,y} \right)}}},{k_{y} - {\frac{2\pi\; n}{\Delta_{\beta}}{T_{2}\left( {x,y} \right)}}}} \right\rbrack}}}}$Using T₁(x,y)=ax+by and T₂(x,y)=cx+dy, the above equation can berewritten as:

${F_{\delta}\left( {k_{x},k_{y}} \right)} = {\frac{1}{\Delta_{x}\Delta_{y}}{\sum\limits_{m = {- \infty}}^{\infty}{\sum\limits_{n = {- \infty}}^{\infty}{F\left\lbrack {{k_{x} - k_{xmn}},{k_{y} - k_{ymn}}} \right\rbrack}}}}$where

$k_{xmn} = {{\frac{2\pi\;{ma}}{\Delta_{\alpha}} + {\frac{2\pi\;{nc}}{\Delta_{\beta}}\mspace{14mu}{and}\mspace{14mu} k_{ymn}}} = {\frac{2\pi\;{mb}}{\Delta_{\alpha}} + {\frac{2\pi\;{nd}}{\Delta_{\beta}}.}}}$The repeated positions of this Fourier transform are at (k_(xmn),k_(ymn)) for m=(−∞,∞) and n=(−∞,∞).

Spectral Property of CFA Green Channel with Two-Dimensional HexagonalSampling—In the case of CFA green channel, the hexagonal sampling hasthe following linear transformation:

$\begin{bmatrix}\alpha \\\beta\end{bmatrix} = {\begin{bmatrix}{T_{1}\left( {x,y} \right)} \\{T_{2}\left( {x,y} \right)}\end{bmatrix} = {\begin{bmatrix}{{ax} + {by}} \\{{cx} + {dy}}\end{bmatrix} = {{\begin{bmatrix}1 & 1 \\{- 1} & 1\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}}.}}}$That is, a=1, b=1, c=−1, and d=1. FIG. 11 illustrates the relationshipbetween two coordinate planes, (α,β) and (x,y). Let the sampling spacesin (x,y) plane be Δ_(x)=1 and Δ_(y)=1, then the sampling spaces in (α,β)plane are Δ_(α)=√{square root over (2)} and Δ_(β)=√{square root over(2)}. The repeated positions of the Fourier transform F_(δ)(k_(x),k_(y))are at

$\left( {k_{xmn},k_{ymn}} \right) = \left( {{\frac{2\pi\; m}{\sqrt{2}} - \frac{2\pi\; n}{\sqrt{2}}},{\frac{2\pi\; m}{\sqrt{2}} + \frac{2\pi\; n}{\sqrt{2}}}} \right)$for m=(−∞,∞) and n=(−∞,∞).

FIG. 11 illustrates a 2-D hexagonal sampling grid. FIG. 12 illustratesthe Fourier transform of F_(δ)(k_(x),k_(y)), which is the repeatedversion of the continuous signal F(k_(x),k_(y)). The distances betweenadjacent two closest repeated spectra in both directions are

$k_{x\; 0} = {{\frac{2\pi}{\Delta_{x}\sqrt{2}}\mspace{14mu}{and}\mspace{14mu} k_{y\; 0}} = {\frac{2\pi}{\Delta_{y}\sqrt{2}}.}}$The cutoff frequency is designed as

$k_{{xs}\; 0} = {{\frac{2\pi}{2\sqrt{2}\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{{ys}\; 0}} = \frac{2\pi}{2\sqrt{2}\Delta_{y}}}$in spatial frequency domain, and

$\rho_{s\; 0} = {\frac{2\pi}{2\sqrt{2}}\rho_{0}}$in the radial spatial frequency as shown in FIG. 12.

Reconstructing CFA Green Channel and Red/Blue Channel—The spectralproperty of the hexagonal sampled data provides the basis forreconstructing the original signal to avoid nonlinear aliasing. FIG. 13is a schematic diagram of sequence of steps representing an algorithmfor reconstructing CFA green channel. The original green signalƒ_(g)(x,y) is passed through an imaging system with the CFA (Box 610) tobecome the hexagonal sampled signal ƒ_(gδ)(x,y). The cutoff frequencyfor the green channel is designed as

$k_{{{xs}\; 0},G} = {{\frac{2\pi}{2\sqrt{2}\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{{{ys}\; 0},G}} = \frac{2\pi}{2\sqrt{2}\Delta_{y}}}$in spatial frequency domain, and

$\rho_{{s\; 0},G} = {\frac{2\pi}{2\sqrt{2}}\rho_{0}}$in the radial spatial frequency (Box 630) based on the spectrum propertyof the CFA green signal. A low-pass filtering W_(g)(k_(x),k_(y)) withthe cutoff frequency ρ_(s0,G) (Box 640) is applied to the Fouriertransform F_(gδ)(k_(x),k_(y)) (Box 620) to generate F_(gL)(k_(x),k_(y)).That is,F _(gL)(k _(x) ,k _(y))=F _(gδ)(k _(x) ,k _(y))W _(g)(k _(x) ,k _(y)).Then the inverse Fourier transform of the low-passed signal (Box 650),F_(gL)(k_(x),k_(y)), is denoted as {tilde over (ƒ)}_(g)(x,y), which isthe reconstructed CFA green signal. The low-pass filter is obtained by aFourier windowing method described in S. S. Young, “Alias-free imagesubsampling using Fourier-based windowing methods,” Optical Engineering,Vol. 43, No. 4, pp. 843-855, April 2004, hereby incorporated byreference.

Reconstructing red or blue channel is similar to the procedure ofreconstructing the green channel except the cutoff frequency for thelow-pass filter is designed as

$k_{{{xs}\; 0},C} = {{\frac{2\pi}{4\Delta_{x}}\mspace{14mu}{and}\mspace{14mu} k_{{{ys}\; 0},C}} = \frac{2\pi}{4\Delta_{y}}}$in spatial frequency domain,

$\rho_{{s\; 0},\; C} = {\frac{2\pi}{4}\rho_{0}}$in radial frequency domain, where C for red or blue channel.Example of Reconstructing Green and Red/Blue Channels

An illustration of reconstructing green and red/blue channels is shownin FIGS. 14-23. In this example, we use a gray scale image to simulate ahexagonal sampled green image and a rectangular subsampled red or blueimage. The subsampled red or blue image is decimated/subsampled by 2from the original image.

The original gray scale image of a nature scene is shown in FIG. 14. Itsspectrum is shown in FIG. 15. Note that in FIG. 15 the spectrum iscentered around the low frequencies. The subsampled rectangular grid forred or blue image which is subsampled by 2 (every other pixel issampled) from the original image is illustrated in FIG. 16. FIG. 17shows the red or blue sampled image. Its spectrum is shown in FIG. 18,in which the repeated versions of the spectrum of the original image arepresented in the rectangular sampling fashion. FIG. 18 is the spectrumfor the red and blue sampled image. Note that by comparing FIG. 18 withFIG. 10, one can see that in FIG. 18 the spectrum is repeated at ninelocations around the perimeter correlating to the repeated versions ofthe original spectrum shown FIG. 10. The reconstructing red or bluesampled image is shown in FIG. 19.

The hexagonal sampling grid for the green image is shown in FIG. 20.FIG. 21 shows the hexagonal sampled green image. Its spectrum is shownin FIG. 22, in which the repeated versions of the spectrum of theoriginal image are presented in the hexagonal sampling fashion. Notethat by comparing FIG. 22 to FIG. 12, one can see repeated versions inthe hexagonal sampling pattern as shown in FIG. 12. The reconstructinggreen sampled image is shown in FIG. 23.

Gross Translation Estimation for Color Filter Array Image Sequences

The gross translation estimation (FIG. 7 Box 214) as described in theforegoing and in U.S. patent application Ser. No. 12/576,132, “Systemand Method of Super-Resolution Imaging from a Sequence of Translated andRotated Low-Resolution Images,” filed Oct. 8, 2009, S. S. Young isillustrated in FIG. 3. The gross translation estimation is applied tothe reconstructed green channel images in the preferred embodiment. Thenthe estimated translations are assigned to the red and blue channels(Box 240). If other CFA pattern is used, the gross translationestimation is applied to the reconstructing luminance channel. Then theestimated translations are assigned to two other chrominance channels.In the alternative embodiment, depicted schematically in FIG. 8, thegross translation estimation is applied to three color channel images,respectively (Box 320, 340, 360). One may also use the estimatedtranslations in other manners; e.g., the averages of the estimatedtranslations for three color channels, respectively, are used for allthree color channels.

SUB-PIXEL ROTATION AND TRANSLATION ESTIMATION FOR COLOR FILTER ARRAYIMAGE SEQUENCES The sub-pixel translation and rotation estimation (Box216) as described in the foregoing and in U.S. patent application Ser.No. 12/576,132, “System and Method of Super-Resolution Imaging from aSequence of Translated and Rotated Low-Resolution Images,” filed Oct. 8,2009, S. S. Young is illustrated in FIGS. 2, 4-6. The sub-pixeltranslation and rotation estimation is applied to the reconstructedgreen channel images in the preferred embodiment. Then the estimatedsub-pixel translations and rotations are assigned to the red and bluechannels (Box 260). If other CFA pattern is used, the sub-pixeltranslation and rotation estimation is applied to the reconstructingluminance channel. Then the estimated translations and rotations areassigned to two other chrominance channels. In the alternativeembodiment, the sub-pixel translation and rotation estimation is appliedto three color channel images, respectively. One may also use theaverages of the estimated translations and rotations for three colorchannels, respectively, for all three color channels, as used by others.

Error-Energy Reduction Algorithm

Error-Energy Reduction Algorithm for CFA Input Images—An error-energyreduction algorithm for CFA images is derived to reconstruct thehigh-resolution alias-free output image that is a three-channel fullcolor image containing red, green and blue channels. A primary featureof this proposed error-energy reduction algorithm for CFA images isthat, for each color channel, we treat the spatial samples fromlow-resolution images that possess unknown and irregular (uncontrolled)sub-pixel shifts and/or rotations as a set of constraints to populate anover-sampled (sampled above the desired output bandwidth) processingarray. The estimated sub-pixel shifted and rotated locations of thesesamples and their values constitute a spatial domain constraint.Furthermore, the bandwidth of the alias-free image (or the sensorimposed bandwidth) is the criterion used as a spatial frequency domainconstraint on the over-sampled processing array.

In an error-energy reduction algorithm, the high-resolution image isreconstructed by removing aliasing from low resolution images. Thehigh-resolution image is reconstructed by applying non-uniforminterpolation using the constraints both in the spatial and spatialfrequency domains. The algorithm is based on the concept that theerror-energy is reduced by imposing both spatial and spatial frequencydomain constraints: the samples from the low-resolution images; and thebandwidth of the high-resolution, alias-free output image.

In accordance with the error-energy reduction algorithm of the presentinvention, spatial samples from low resolution images that possessunknown and irregular (uncontrolled) sub-pixel shifts and rotations aretreated as a set of constraints to populate an over-sampled (sampledabove the desired output bandwidth) processing array. The estimatedsub-pixel locations of these samples and their values constitute aspatial domain constraint. Furthermore, the bandwidth of the alias-freeimage (or the sensor imposed bandwidth) is the criterion used as aspatial frequency domain constraint on the over-sampled processingarray. One may also incorporate other spatial or spatial frequencydomain constraints that have been used by others; e.g., positivity ofthe image in the spatial domain. Since each color channel presents aunique spatial and spatial frequency properties as described earlier,the spatial and spatial frequency domain constraints are designedaccordingly.

Image acquisition model—FIG. 24 shows the system model for acquiring anundersampled image in a one-dimensional case. Let ƒ(x) be the idealtarget signature that is interrogated by the sensor. The measured targetsignature g(x) by the sensor is modeled as the output of a Linear ShiftInvariant (LSI) system whose impulse response is the sensor's pointspread function (PSF), h(x); this is also known as the sensor blurringfunction. The relationship between the measured target signature and theoriginal target signature in the spatial frequency k_(x) domain is:G(k _(x))=F(k _(x))H(k _(x))where G(k_(x)), F(k_(x)), and H(k_(x)) are the Fourier transforms ofg(x), ƒ(x), and h(x), respectively.

FIG. 25 shows the factors that dictate the bandwidth of the measuredtarget signature. The bandwidth of the sensor's PSF (point spreadfunction) is fixed and is determined by the support of the transferfunction H(k_(x)). The bandwidth of the ideal target signature dependson, e.g., the range of the target in FLIR (Forward-Looking Infrared),radar, sonar, visible light, etc. For a target at the short range, moredetails of the target are observable; in this case, the ideal targetsignature bandwidth is relatively large. For a target at the long range,the ideal target signature bandwidth is smaller. Two lines (one dottedand one solid) in FIG. 25 illustrate the bandwidths of the sensor andthe ideal target signature. The wider (dotted) one could be thebandwidth of the target or the sensor, or visa-versa. The outputbandwidth of the measured target signature is the minimum bandwidth ofthe ideal target signature and the sensor, that is,B _(o)=min(B _(t) ,B _(s))where B_(o), B_(t) and B_(s) are the bandwidths of the output targetsignature, the ideal target signature and the sensor PSF, respectively.The proposed super-resolution image reconstruction algorithm cannotproduce an image whose bandwidth exceeds B_(o), because the de-aliasingapproach cannot recover the bandwidth beyond the sensor's diffractionlimit.

Assuming the measurement system stores the incoming images at the samplespacing of Δ_(x), the measured target signature, g(x), is then sampledat the integer multiples of Δ_(x), e.g., x_(n)=nΔ_(x),n=0, ±1, ±2, . . ., ±N, to obtain the undersampled (aliased) image; the resultant can bemodeled as the following delta-sampled signal in the spatial domain (seeFIG. 24);

${g_{\delta}(x)} = {{\sum\limits_{n = {- N}}^{N}{{g(x)}{\delta\left( {x - x_{n}} \right)}}} = {{g(x)}{\sum\limits_{n = {- N}}^{N}{{\delta\left( {x - x_{n}} \right)}.}}}}$

The Fourier transform of the above delta-sampled signal is

${G_{\delta}\left( k_{x} \right)} = {\frac{1}{\Delta_{x}}{\sum\limits_{l = {- N}}^{N}{{G\left( {k_{x} - \frac{2\pi\; l}{\Delta_{x}}} \right)}.}}}$

An acquired image is aliased when

$\Delta_{x} > {\frac{2\pi}{B_{o}}\mspace{14mu}{\left( {{Nyquist}\mspace{14mu}{sample}\mspace{14mu}{spacing}} \right).}}$

As mentioned previously, many low resolution sensors possess a spatialsample spacing Δ_(x) that violates the Nyquist rate. The emphasis of thesuper-resolution image reconstruction algorithm in a preferredembodiment is to de-alias a sequence of undersampled aliased images toobtain the alias-free or a super-resolved image. Again it should beemphasized that the resolution of the output alias-free image is limitedby the minimum bandwidth of the sensor and the ideal target signature.

One of the considerations in super-resolution image reconstruction isthe number of the undersampled images that are required to recover thealias-free image. For monochrominance images, let the sampling space ofthe original undersampled images be Δ_(x). From an information-theorypoint of view, the sampling space of the output high-resolution imagethat is reconstructed from k frames (with distinct sub-pixel shifts) is

$\frac{\Delta_{x}}{k}$in one-dimensional imaging systems, and

$\left( {\frac{\Delta_{x}}{\sqrt{k}},\frac{\Delta_{y}}{\sqrt{k}}} \right)$in two-dimensional imaging systems.

A numerical example is given in the following. An image array of 100×100is initially acquired to represent a scene. The total number of pixelsis 10,000. If the resolution of the image is desired to be 4 times ineach direction, the image array of 400×400 is required. The total numberof pixels for the high-resolution image is 160,000. That means that itrequires 16 low-resolution images to achieve the resolution improvementfactor of 4. Therefore, the number of input low-resolution image framesis the square of the resolution improvement factor. In other words, theresolution improvement factor is the square root of the number of theinput images.

Now consider a CFA image, an image array of n_(x)*n_(y) is initiallyacquired to represent a scene. The total number of pixels isN=n_(x)*n_(y). The total number pixels in the green channel isN_(G)=N/2, while the total number of pixels in the blue or red channelis N_(C)=N/4, where C represents for red or blue channel. When k framesof CFA images are acquired to reconstruct the high-resolution full colorimage, the total number of pixels of the green channel of the outputhigh-resolution image is

${N_{G\_{SR}} = {\frac{N}{2}k}},$the one for the blue or red channel is

${N_{C\_{SR}} = {\frac{N}{4}k}},$where C represents for red or blue channel. Therefore, the resolutionimprovement factor in the green channel is √{square root over (k/2)} inboth x and y directions, and the one in the red or blue channel √{squareroot over (k/4)}. That is, for CFA images, let the sampling space of theoriginal undersampled images be Δ_(x). From an information-theory pointof view, the sampling space of the output high-resolution full colorimage that is reconstructed from k frames (with distinct sub-pixelshifts) is

$\left( {\frac{\Delta_{x}}{\sqrt{k/2}},\frac{\Delta_{y}}{\sqrt{k/2}}} \right)$in the green channel,

$\left( {\frac{\Delta_{x}}{\sqrt{k/4}},\frac{\Delta_{y}}{\sqrt{k/4}}} \right)$in the red or blue channel in two-dimensional imaging systems. Inanother words, the resolution improvement factor of the green channel is√{square root over (k/2)} and the one of the red or blue channel is√{square root over (k/4)}.

Theoretically, more frames should provide more information and, thus,larger bandwidth and/or higher resolution. However, the super-resolutionimage quality does not improve when the number of processed undersampledimages exceeds a certain value. This is due to the fact that thebandwidth recovery is limited by the minimum of the bandwidths of thesensor or ideal target signature.

Implementation of Error-Energy Reduction Algorithm for CFA Images—Foreach color channel, the bandwidth of the input undersampled (aliased)images is designated B_(i). The bandwidth of the alias-free(high-resolution) output image for that channel is denoted as B_(o). Inorder to recover the desired (alias-free) bandwidth, it is important toselect a processing array with a sample spacing smaller than the samplespacing of the desired high-resolution image. In this case, the 2D FFTof the processing array yields a spectral coverage B_(p) that is largerthan the bandwidth of the output image. FIG. 26 illustrates thisphenomenon in the 2D spatial frequency domain for one color channel.

The high-resolution alias-free image on the processing array (grid) isdesignated as p_(D)(x,y) where Dε{G,C}, G for the green channel andCε{R,B} for the red or blue channel. The low-resolution (aliased)snap-shots only provide a sub-set of samples of this image. The locationof each aliased image on the processing grid is dictated by itscorresponding sub-pixel shifts and rotations in the (x,y) domain.

In the following, the error-energy reduction algorithm is described forthe CFA input images containing translational shifts and/or rotationaldifferences. FIG. 27 shows a schematic representation of theerror-energy reduction algorithm for CFA input image sequencestransformed by translations and/or rotations.

Inputs to the error energy reduction algorithm are a sequence oflow-resolution CFA input images. In the preferred embodiment, theestimated gross shifts and sub-pixel shifts and/or rotations of eachimage with respect to the reference image are obtained for each colorchannel; usually the first image is chosen as the reference image.Denote ƒ_(l)(u_(i),v_(j)) to be L CFA images from the input imagesequence, where l=1, 2, . . . , L, i=1, 2, . . . , I, j=1, 2, . . . , J;I×J is the dimension of the input image. After the transformationparameters ({circumflex over (θ)}_(l),{circumflex over (x)}_(sl),ŷ_(sl))are estimated for all L images, the transform coordinates are calculatedfrom the estimated sub-pixel shifts and rotation angles as follows:

$\begin{bmatrix}{\hat{x}}_{ijl} \\{\hat{y}}_{ijl}\end{bmatrix} = {{\begin{bmatrix}{\cos\;{\hat{\theta}}_{l}} & {\sin\;{\hat{\theta}}_{l}} \\{{- \sin}\;{\hat{\theta}}_{l}} & {\cos\;{\hat{\theta}}_{l}}\end{bmatrix}\begin{bmatrix}u_{i} \\v_{j}\end{bmatrix}} + \begin{bmatrix}{\hat{x}}_{sl} \\{\hat{y}}_{sl}\end{bmatrix}}$In the preferred embodiment, the transform coordinates are the same forall three color channels.

In the initializing step (Box 402), three processing arrays areinitialized for three color channels by populating each grid using theinput images according to the estimates of shifts and rotation angles,respectively. Each processing array is selected with a sample spacingthat is smaller than the sample spacing of the desired high-resolutionimage.

In this procedure, the available CFA image values (from multiplesnapshots) for three color channels are assigned to each shifted androtated grid location of three processing arrays, respectively. At theother grid locations of each processing array, zero values are assigned.This can be written as

${p_{1,D}\left( {X_{m},Y_{n}} \right)} = \left\{ \begin{matrix}{f_{l,D}\left( {{\hat{x}}_{ijl},{\hat{y}}_{ijl}} \right)} \\0\end{matrix} \right.$where Dε{G,C}, G for the green channel and Cε{R,B} for the red or bluechannel, (X_(m),Y_(n)) is the grid location that is closest to thetransformation coordinates [{circumflex over (x)}_(ijl), ŷ_(ijl)], andm=1, 2, . . . , M, n=1, 2, . . . , N; M×N is the dimension of theprocessing grid. The input green channel ƒ_(1,G) follows the samplingpattern of the hexagonal sampling shown in FIG. 20, and the input red orblue channel ƒ_(1,C), follows the sampling pattern of the rectangularsampling shown in FIG. 16.

In Box 404, a 2-D Fourier transform P_(1,D)(k_(x),k_(y)) is preformed onthe processing array, where Dε{G,C}, G for the green channel and Cε{R,B}for the red or blue channel. Its spectrum has a wider bandwidth than thetrue (desired) output bandwidth. Therefore, the spatial frequency domainconstraint (Box 408) is applied, that is, replacing zeros outside thedesired bandwidth.

In this step, the desired bandwidth for the green channel and the onefor red or blue channel are determined first (Box 406). Based on thespectral properties of the hexagonal sampling and the resolutionimprovement factor for the green channel when L CFA images are acquired,the desired bandwidth for the green channel is obtained as

$\left( {B_{{Ox},G},B_{{Oy},G}} \right) = \left( {{\frac{2\pi}{2\sqrt{2}\Delta_{x}}\sqrt{L/2}},{\frac{2\pi}{2\sqrt{2}\Delta_{y}}\sqrt{L/2}}} \right)$in the spatial frequency domain. From the spectral properties of therectangular sampling and the resolution improvement factor for the redor blue channel when L CFA images acquired, the desired bandwidth forthe red or blue channel is obtained as

$\left( {B_{{Ox},C},B_{{Oy},C}} \right) = \left( {{\frac{2\pi}{4\Delta_{x}}\sqrt{L/4}},{\frac{2\pi}{4\Delta\; y}\sqrt{L/4}}} \right)$in the spatial frequency domain.The functionsP _(2,D)(k _(x) ,k _(y))=P _(1,D)(k _(x) ,k _(y))·W _(p,D)(k _(x) ,k_(y))is formed where W_(p,D)(k_(x),k_(y)) is a windowing function ofrealizing the spatial frequency domain constraint using(B_(Ox,G),B_(Oy,G)) for the green channel and (B_(Ox,c), B_(Oy,C)) forthe red or blue channel, respectively. In Box 410, the inverse 2-DFourier transform p_(2,D)(x,y) of the resultant arrayP_(2,D)(k_(x),k_(y)) is performed for all three color channels,respectively.

The resultant inverse array is not a true image because the image valuesat the known grid locations are not equal to the original image values.Then the spatial domain constraint is applied for all three colorchannels (Box 412), that is, replacing those image values at the knowngrid locations with the known original image values.

In this step of applying spatial domain constraints using shifts androtations (Box 412), we have the function

${p_{{k + 1},D}\left( {X_{m},Y_{n}} \right)} = \left\{ \begin{matrix}{f_{l,D}\left( {{\hat{x}}_{ijl},{\hat{y}}_{ijl}} \right)} \\{p_{k,D}\left( {X_{m},Y_{n}} \right)}\end{matrix} \right.$to form the current iteration output image p_(k+1,D)(X_(m),Y_(n)) fromthe previous iteration output image p_(k,D)(X_(m),Y_(n)) where Dε{G,C},G for the green channel and Cε{R,B} for the red or blue channel. Thatis, the known grid locations are replaced with the known original imagevalues and other grid locations are kept as the previous iterationoutput image values.

Similarly, the use of the available information in the spatial andspatial frequency domains results in a reduction of the energy of theerror at each iteration step. At the odd steps of the iteration, theerror for each of three color channel is defined by

$e_{{{2k} + 1},D} = {\sum\limits_{({m,n})}\left\lbrack {{p_{{{2k} + 1},D}\left( {X_{m},Y_{n}} \right)} - {p_{{{2k} - 1},D}\left( {X_{m},Y_{n}} \right)}} \right\rbrack^{2}}$where Dε{G,C}, G for the green channel and Cε{R,B} for the red or bluechannel.

In Box 414, the condition of the error-energy reduction is examined bydefining the following ratio:

${S\; N\; R_{{{2k} + 1},D}} = {10{\log_{10}\left( \frac{\sum\limits_{({m,n})}\left\lbrack {p_{{{2k} + 1},D}\left( {X_{m},Y_{n}} \right)} \right\rbrack^{2}}{\sum\limits_{({m,n})}\left\lbrack {{p_{{{2k} + 1},D}\left( {X_{m},Y_{n}} \right)} - {p_{{{2k} - 1},D}\left( {X_{m},Y_{n}} \right)}} \right\rbrack^{2}} \right)}}$where Dε{G,C}, G for the green channel and Cε{R,B} for the red or bluechannel. If SNR_(2k+1,D)<SNR_(max,D) (where SNR_(max,D) is apre-assigned threshold value), and the maximum iteration is not reached(Box 416), the iteration continues. If SNR_(2k+1,D)>SNR_(max,D), thatis, the stopping criterion is satisfied, the iteration is terminated.Before the final super-resolution image is generated, the bandwidth ofthe output image is reduced (Box 418) from B_(p) to B_(o) using theFourier windowing method (see S. S. Young 2004) for the green, red, andblue channel, respectively. Then the final super-resolved full colorimage with a desired bandwidth is saved for the output (Box 420).

EXAMPLE

An illustration of super-resolving images from CFA low-resolutionsequences is shown in FIGS. 28-29. FIG. 28 shows an example of scene ofa license plate. One of original CFA input images is shown in the leftcolumn. The super-resolved three channels including the red, green andblue channels are shown in the middle column. These three images can beused to display as a color image. In order to show the effectiveness ofthe super-resolved full color image, the super-resolved luminance imageis shown in the right column. The conversion from RGB to luminance ispresented in the first row of the following matrix:

$T = \begin{bmatrix}0.2989 & 0.5870 & 0.1140 \\0.5959 & {- 0.2744} & {- 0.3216} \\0.2115 & {- 0.5229} & 0.3114\end{bmatrix}$In order to protect the license plate owner's privacy, part of thelicense plate is covered.

Another example of scene of a car emblem is illustrated in FIG. 29. Byobserving FIG. 29, it can be seen that the super-resolved image shows asignificant improvement from the CFA low-resolution images by exhibitingthe detailed information especially in the digits and letters in thelicense plate and the car emblem. This illustrates that thesuper-resolved image is obtained even under the condition that a colorfilter array image sequence is acquired.

FIG. 30 depicts a high level block diagram of a general purpose computersuitable for use in performing the functions described herein, includingthe steps shown in the block diagrams, schematic representations, and/orflowcharts. As depicted in FIG. 30, the system 500 includes a processorelement 502 (e.g., a CPU) for controlling the overall function of thesystem 500. Processor 502 operates in accordance with stored computerprogram code, which is stored in memory 504. Memory 504 represents anytype of computer readable medium and may include, for example, RAM, ROM,optical disk, magnetic disk, or a combination of these media. Theprocessor 502 executes the computer program code in memory 504 in orderto control the functioning of the system 500. Processor 502 is alsoconnected to network interface 505, which transmits and receives networkdata packets. Also included are various input/output devices 506 (e.g.,storage devices, including but not limited to, a tape drive, a floppydrive, a hard disk drive or compact disk drive, a receiver, atransmitter, a speaker, a display, a speech synthesizer, an output port,and a user input device (such as a keyboard, a keypad, a mouse and thelike).

Although various preferred embodiments of the present invention havebeen described herein in detail to provide for complete and cleardisclosure, it will be appreciated by those skilled in the art, thatvariations may be made thereto without departing from the spirit of theinvention.

As used herein, the terminology “subject” means: an area, a scene, anobject or objects, a landscape, overhead view of land or an object orobjects, or a combination thereof.

As used herein, the terminology “frame” means: a picture, an image orone of the successive pictures on a strip of film or video.

As used herein, the terminology “process” means: an algorithm, software,subroutine, computer program, or methodology.

As used herein, the terminology “algorithm” means: sequence of stepsusing computer software, process, software, subroutine, computerprogram, or methodology.

As used herein, the terminology “demosaicing process” is a digital imageprocess of reconstructing a full color image from incomplete colorsamples output from an image sensor or camera overlaid with a colorfilter array (CFA). Demosaicing or demosaicing is also referred to asCFA interpolation or color reconstruction.

As used herein, the terminology “image sensor” means: a camera, chargecoupled device (CCD), video device, spatial sensor, or range sensor. Theimage sensor may comprise a device having a shutter controlled aperturethat, when opened, admits light enabling an object to be focused,usually by means of a lens, onto a surface, thereby producing aphotographic image OR a device in which the picture is formed before itis changed into electric impulses.

The terminology “transformation” as used herein (not to be confused withthe terminology Fourier transform) refers to changes induced by movementof the image capture device, including translation and rotation.

The terminology “processor” or “image processor” as used in thefollowing claims includes a computer, multiprocessor, CPU, minicomputer,microprocessor or any machine similar to a computer or processor whichis capable of processing algorithms.

The terminology “luminescence” or “luminescence contrast” refers to thedifference in luminosity (or lightness) of an object as compared to thebackground. A more detailed description of the human visual reaction tovarious levels of contrast is found at NASA's Applied Color Science sitehttp://colorusage.arc.nasa.gov/design_lum_(—)1.php, hereby incorporatedby reference. In general, the primary colors are been given luminancesby design, to achieve white alignment of the monitor. Early designers ofcomputer displays decided that equal R, G, and B digital codes shouldproduce a particular neutral chromaticity, the “alignment white” of themonitor (usually 6500K or 5300K). Once the white alignment color wasdecided and the chromaticities of the three primaries were chosen themaximum luminances of the primaries were set so that luminances of theprimaries would mix to give the alignment white. For example, the humaneye is more sensitive to the color green. For example, the color greenis used in the color array of FIG. 1 to a greater extent than red orblue, so that green is designated as the luminescent color. In graphicsthat have the appearance of surfaces (like this page, for example) itcan be useful to describe luminance contrast in terms of the differencebetween the lightness of the symbol or text and its background, relativeto the display's white-point (maximum) luminance.

As used herein, the terminology “spatial anti-aliasing” refers to atechnique in which distortion artifacts (also refer to as aliasing) areminimized when creating a high-resolution image at a lower resolution.Anti-aliasing relates to the process of removing signal components of afrequency higher than is able to be properly resolved by the opticalrecording or sampling device.

The terminology “operations” as used in the following claims includessteps, a series of operations, actions, processes, subprocesses, acts,functions, and/or subroutines.

As used herein the terminology “color channel” is used in the context ofa grayscale image of the same size as a color image, made of just one ofthe primary colors. For example, an image from a standard digital cameragenerally has green, red, and blue channels. A grayscale image has justone channel.

The terminology “full color” as used herein includes three primary colorchannels where three colors are utilized to produce the color image orfour or more primary color channels where four or more colors are usedto produce the color image.

It should be emphasized that the above-described embodiments are merelypossible examples of implementations. Many variations and modificationsmay be made to the above-described embodiments. All such modificationsand variations are intended to be included herein within the scope ofthe disclosure and protected by the following claims. The invention hasbeen described in detail with particular reference to certain preferredembodiments thereof, but it will be understood that variations andmodifications can be effected within the spirit and scope of theinvention.

What is claimed is:
 1. A system for improving the quality of colorimages by combining the content of a plurality of frames of the samesubject comprising: at least one processor; the at least one processoroperatively connected to a memory for storing a plurality of frames of asubject; the frames containing imperfect images of the subject due beingaffected by translation and/or rotational movement; the at least oneprocessor operating to align and combine the content of plurality offrames of the subject into a combined multicolored image; the at leastone processor operating to convert at least two multicolored frames intomonochromatic predetermined color frames; the at least one processoroperating to estimate the optimal orientation by using the at least twomonochromatic frames and by estimating optimal alignment of the twomonochromatic frames at the frame and subpixel levels; at the framelevel, the at least one processor operating to perform a gross shiftprocess in which the gross shift translation of one monochromaticpredetermined color frame is determined relative to anothermonochromatic predetermined color frame; at the subpixel level, the atleast one processor operating to perform a subpixel shift processutilizing a correlation method to determine the translational and/orrotational differences of one monochromatic predetermined color frame tothe other monochromatic predetermined color frame to estimate sub-pixelshifts and/or rotations between the frames; the at least one processorcombining the at least two multicolored frames into a multicolored colorimage based upon the results of the monochromatic gross shift andsubpixel shift processes; and checking to determine whether theresolution of the resulting combined multicolored image is of sufficientresolution; the checking process operating to determine the validity ofthe gross shift and subpixel shift processes to produce at least onehigh-resolution multicolored image.
 2. The system of claim 1 wherein theprocess for converting at least two frames to predetermined color framescomprises changing at least two multicolored frames into monochromaticpredetermined color frames by substituting pixels of the predeterminedcolor for pixels which are not of the predetermined color to createmonochromatic predetermined color frames.
 3. The system of claim 2 wherethe process of substituting pixels of the predetermined color for pixelswhich are not of the predetermined color comprises keeping the pixels ofpredetermined color and zeroing the pixels which are not of thepredetermined color for each frame, converting the resultant frame intoa signal, performing a Fourier transform on the signal, determining thecut-off frequency for the signal, passing the signal through low-passwindowing, and performing an inverse Fourier transform on the signal toobtain the predetermined colored pixel values for every pixel locationin the frame and subsequently repeating the process for the plurality offrames to produce a monochromatic color sequence of frames.
 4. Thesystem of claim 3 further comprising creating a monochromatic colorsequence of frames for each of the primary colors in the multicoloredframes and assigning the estimated gross shift and estimated subpixelshift obtained using the predetermined color frame sequence to the othercolor frame sequences to obtain a full color image and wherein the fullcolor image comprises one of (a) green, red, blue or (b) green, red,blue, and white or (c) red, green, blue, and cyan.
 5. The system ofclaim 1 wherein the subpixel shift process comprises estimating therotation and/or translation of one frame relative to the reference frameof one of a plurality of primary colors.
 6. The system of claim 1wherein the plurality of frames of the same subject are an undersampledsequence of frames of a moving subject and/or are subject to jitter. 7.The system of claim 1 wherein the plurality of frames comprise pixels ofthree primary colors and wherein the checking process comprises: a)estimating translation and/or rotation values for three channels; b)initializing a processing arrays for each of the three color channels bypopulating the grid of each processing array using data from arespective one of the three sampled color channels and the translationand rotation estimates for each color channel; c) applying a 2D Fouriertransform to the three processing arrays, respectively, to obtainFourier transformed processing arrays; d) determining spatial frequencydomain constraints for each of the three processing arrays; e) applyingspatial frequency domain constraints to each of the three the Fouriertransformed processing arrays to obtain the constrained spatialfrequency domain processing arrays; f) applying an inverse 2D Fouriertransform to the constrained spatial frequency domain processing arraysto obtain inverse processing arrays for each color; g) applying spatialdomain constraints using the estimated shifts and rotations to the saidinverse processing arrays to obtain the constrained spatial domainprocessing arrays for each color; h) checking the condition; if thestopping criterion is not satisfied, repeating the steps (c) through(g); and if the stopping criterion is satisfied then i) reducing thebandwidth from each of the processing arrays to an output arraybandwidth using the Fourier windowing method; and j) outputting asuper-resolved multicolored image with the desired bandwidth.
 8. Asystem for super-resolving images from color filter array (CFA)low-resolution sequences comprising: at least one processor having aninput for inputting multiple CFA images of a scene with sub-pixeltranslations and rotations; one of the images of the scene being areference image; and at least one memory operatively connected to the atleast one processor, the at least one processor operating to estimatethe optimal orientation by first converting the multicolored frames tomonochromatic frames using a monochromatic color image constructingalgorithm for creating at least one monochromatic color image for eachimage in the input sequence; the at least one processor operating toestimate the shift of the inputted CFA images using monochromatic framesand performing estimates of the amount of shift of the image from oneimage to a reference image at the frame and subpixel levels by using agross translation estimation algorithm for estimating overalltranslations of the at least one monochromatic color image with respectto the reference image on the image processor; a sub-pixel estimationalgorithm for estimating the sub-pixel translations and/or rotations ofat least one monochromatic color image with respect to the referenceimage; and the at least one processor operating to check and validatethe estimates obtained using the gross translation and subpixelprocesses to produce at least one high-resolution full color image. 9.The system of claim 8 wherein the monochromatic color image constructingalgorithm creates a plurality of monochromatic color sequences andwherein the gross translation estimation and subpixel estimationperformed on one of monochromatic color sequences are assigned to othermonochromatic color sequences.
 10. The system of claim 8 wherein theoperations performed by the monochromatic color image constructingalgorithm comprise obtaining the hexagonal sampled image; applying 2-DFourier transform to the hexagonal sampled image to obtain transformedimage; determining cutoff frequency; applying low-pass windowinggenerated from the cutoff frequency to the transformed image to createlow-pass windowed image; and applying inverse 2-D Fourier transform tothe low-pass windowed image.
 11. The system of claim 8 wherein themonochromatic color image constructing algorithm processes at leastthree color channel sequences and the gross translations and sub-pixeltranslations and rotations are estimated for at least three colorchannel sequences.
 12. The system of claim 8 wherein the validating andchecking operations performed by the monochromatic color imageconstructing algorithm comprises obtaining either hexagonal sampled orrectangular sampled images for each monochromatic color, applying 2-DFourier transform to the sampled images to obtain Fourier transformimages; determining cutoff frequencies; applying low-pass windowinggenerated from the cutoff frequencies to the Fourier transformed imagesto obtain low-pass window images; and applying inverse 2-D Fouriertransform to the low-pass windowed images.
 13. The system of claim 12wherein the monochromatic color image constructing algorithm processesat least three color channel sequences i. one color is the luminancechannel and is obtained using hexagonal sampled images; ii. two colorsare the chrominance colors and are obtained using rectangular sampledimages; and wherein the operations performed by determining cutofffrequencies comprise designing different cutoff frequencies for theluminance channel and chrominance channels.
 14. The system of claim 8wherein the validating and checking operations comprise: a) inputtingCFA images and estimated translation and rotation values for three colorchannels; b) initializing a processing arrays for each of the threecolor channels by populating the grid of each processing array usingdata from a respective one of the three sampled color channels and theestimated translation and rotation estimates for each color channel; c)applying a 2D Fourier transform to the three processing arrays,respectively, to obtain Fourier transformed processing arrays; d)determining spatial frequency domain constraints for each of the threeprocessing arrays; e) applying spatial frequency domain constraints toeach of the three the Fourier transformed processing arrays to obtainthe constrained spatial frequency domain processing arrays; f) applyingan inverse 2D Fourier transform to the constrained spatial frequencydomain processing arrays to obtain inverse processing arrays for eachcolor; g) applying spatial domain constraints using the estimated shiftsand rotations to the said inverse processing arrays to obtain theconstrained spatial domain processing arrays for each color; h) checkingthe condition; if the stopping criterion is not satisfied, repeating thesteps (c) through (g); and if the stopping criterion is satisfied theni) reducing the bandwidth from each of the processing arrays to anoutput array bandwidth using the Fourier windowing method; and j)outputting a super-resolved full color image with the desired bandwidth.15. The system of claim 8 wherein the validating and checking operationscomprise applying at least one spatial frequency domain constraint usingthe cutoff frequency designed from the hexagonal sampling properties forthe green channel; and using the cutoff frequency designed from therectangular sampling properties for one of the other color channels. 16.A method of super-resolving CFA images affected by movement includingtranslation and rotation comprising: inputting frames of low-resolutionCFA images of a scene into a processor; at least one of the frames beingaffected by movement; obtaining at least one monochromatic color imagefor each image in the input frames; estimating the gross translationsand sub-pixel shifts and rotations due to the movement using themonochromatic color images; and checking the validity of the estimatedgross translations and subpixel shifts to produce a high-resolution fullcolor output image.
 17. The method of claim 16 wherein the step ofobtaining a monochromatic color image is completed for a plurality ofdifferent color monochromatic colors and the estimated grosstranslations and sub-pixel translations and rotations from the firstmonochromatic channel are assigned to other monochromatic channels. 18.The method of claim 16 wherein the step of obtaining a monochromaticcolor image comprises obtaining the hexagonal sampled images; applying2-D Fourier transform to each hexagonal sampled portion to obtain aFourier transform image; determining cutoff frequency; applying low-passwindowing generated from the cutoff frequency to each Fouriertransformed images to obtain a low pass window image; and applyinginverse 2-D Fourier transform to each low-pass windowed image.
 19. Themethod of claim 16 wherein the steps of checking and validatingoperations comprise: a) inputting CFA images and estimated translationand rotation values for three color channels; b) initializing aprocessing arrays for each of the three color channels by populating thegrid of each processing array using data from a respective one of thethree sampled color channels and the estimated translation and rotationestimates for each color channel; c) applying a 2D Fourier transform tothe three processing arrays, respectively, to obtain Fourier transformedprocessing arrays; d) determining spatial frequency domain constraintsfor each of the three processing arrays; e) applying spatial frequencydomain constraints to each of the three Fourier transformed processingarrays to obtain the constrained spatial frequency domain processingarrays; f) applying an inverse 2D Fourier transform to the constrainedspatial frequency domain processing arrays to obtain inverse processingarrays for each color; g) applying spatial domain constraints using theestimated shifts and rotations to the said inverse processing arrays toobtain the constrained spatial domain processing arrays for each color;h) checking the condition; if the stopping criterion is not satisfied,repeating the steps (c) through (g); and if the stopping criterion issatisfied then i) reducing the bandwidth from each of the processingarrays to an output array bandwidth using the Fourier windowing method;and j) outputting a super-resolved full color image with the desiredbandwidth.
 20. The method of claim 19 wherein the step of initializingprocessing arrays comprise: a) providing the estimated gross translationand sub-pixel shift and rotation of each monochromatic color image withrespect to a reference image; b) calculating the transformed coordinatesfrom the estimated gross translations and sub-pixel shifts and rotationangles for each input monochromatic color image; c) generating three 2Dprocessing arrays with a sample spacing smaller than the desiredhigh-resolution output image, d) assigning the known green monochromaticchannel values to the closest gross translated and sub-pixel shifted androtated grid locations of the predetermined processing array in ahexagonal sampling pattern; e) assigning the known red or blue imagevalues to the closest gross translated and sub-pixel shifted and rotatedgrid locations of the red or blue processing array in a rectangularsampling pattern; and f) assigning zeros to other grid locations. 21.The method of claim 16 wherein the checking operation comprises aniterative error energy reduction algorithm using the transformations ofeach color channel low-resolution image in three upsampled grids toreconstruct the high resolution three channel full color output, theerror-energy reduction algorithm utilizing the low resolution CFA imagesand their transformations of each color channel on the upsampled gridsas the spatial domain constraint and the bandwidth of each color channelas the spatial frequency domain constraint.
 22. The method of claim 19wherein spatial domain constraint is the estimated translations androtations of the low resolution green channel images in a hexagonalsampling pattern and their values; the estimated translations androtations of the low-resolution red or blue channel images in arectangular sampling pattern and their values; and the spatial frequencydomain constraint is the bandwidth determined from cutoff frequencies ofhexagonal sampled green channel image and rectangular sampled red orblue channel images.